reserve f for Function;
reserve n,k,n1 for Nat;
reserve r,p for Real;
reserve x,y,z for object;
reserve seq,seq1,seq2,seq3,seq9,seq19 for Real_Sequence;

theorem Th13:
  (seq1 + seq2) + seq3 = seq1 + (seq2 + seq3)
proof
  now
    let n be Element of NAT;
    thus ((seq1+seq2)+seq3).n=(seq1+seq2).n+ seq3.n by Th7
      .=seq1.n+seq2.n+seq3.n by Th7
      .=seq1.n+(seq2.n+seq3.n)
      .=seq1.n+(seq2+seq3).n by Th7
      .=(seq1+(seq2+seq3)).n by Th7;
  end;
  hence thesis by FUNCT_2:63;
end;
